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As we know, AMX 10 RC is a very special wheeled AFV for it's skid steering system.
It can perform neutral steer, a.k.a. pivot steer or steer in place.

The transmission case is similar to some tank transmission.

The driver uses steering levers instead of a steerig wheel, and it's almost identical with those on AMX 10P tracked AFV.

↑wheeled AMX 10 RC
↓tracked AMX 10 P

It even had a variant running on tracks:

As for steering principle, there are different descriptions.

Some info says AMX 10 RC use a triple-differential system, like some British tanks and AFVs. A similar skid-steering wheeled vehicle, TV1000 "Rhino" also use steering levers instead of steering wheel, and it have triple-differential system.

When I looked for historical information, I found that French invented the first double-differential steering system for wheeled vehicle since 1898. And French tanks used double differential system since 1920s:

I always knew the Pz 35 used a compressed air system for changing gears, this has been written about in several books and said to have been rendered inoperable by the Russian Winter. I didn't know until watching the new "Matilda Diaries" episode on youtube that the Matilda II infantry tank also used compressed air to change gears. Anyone know if any other WW2 tanks used compressed air systems? Any records of the Soviets having issues with this system in their Lend Lease Matilda tanks in the winter?

Tank design is often conceptualized as a balance between mobility, protection and firepower. This is, at best, a messy and imprecise conceptualization. It is messy because these three traits cannot be completely separated from each other. An APC, for example, that provides basic protection against small arms fire and shell fragments is effectively more mobile than an open-topped vehicle because the APC can traverse areas swept by artillery fires that are closed off entirely to the open-topped vehicle. It is an imprecise conceptualization because broad ideas like "mobility" are very complex in practice. The M1 Abrams burns more fuel than the Leo 2, but the Leo 2 requires diesel fuel, while the omnivorous AGT-1500 will run happily on anything liquid and flammable. Which has better strategic mobility? Soviet rail gauge was slightly wider than Western European standard; 3.32 vs 3.15 meters. But Soviet tanks in the Cold War were generally kept lighter and smaller, and had to be in order to be moved in large numbers on a rail and road network that was not as robust as that further west. So if NATO and the Warsaw Pact had switched tanks in the late 1950s, they would both have downgraded the strategic mobility of their forces, as the Soviet tanks would be slightly too wide for unrestricted movement on rails in the free world, and the NATO tanks would have demanded more logistical support per tank than evil atheist commie formations were designed to provide.

So instead of wading into a deep and subtle subject, I am going to write about something that is extremely simple and easy to describe in mathematical terms; the top speed of a tank moving in a straight line. Because it is so simple and straightforward to understand, it is also nearly meaningless in terms of the combat performance of a tank.

In short, the top speed of a tank is limited by three things; the gear ratio limit, the power limit and the suspension limit. The tank's maximum speed will be whichever of these limits is the lowest on a given terrain. The top speed of a tank is of limited significance, even from a tactical perspective, because the tank's ability to exploit its top speed is constrained by other factors. A high top speed, however, looks great on sales brochures, and there are examples of tanks that were designed with pointlessly high top speeds in order to overawe people who needed impressing.

When this baby hits 88 miles per hour, you're going to see some serious shit.

The Gear Ratio Limit

Every engine has a maximum speed at which it can turn. Often, the engine is artificially governed to a maximum speed slightly less than what it is mechanically capable of in order to reduce wear. Additionally, most piston engines develop their maximum power at slightly less than their maximum speed due to valve timing issues:

A typical power/speed relationship for an Otto Cycle engine. Otto Cycle engines are primitive devices that are only used when the Brayton Cycle Master Race is unavailable.

Most tanks have predominantly or purely mechanical drivetrains, which exchange rotational speed for torque by easily measurable ratios. The maximum rotational speed of the engine, multiplied by the gear ratio of the highest gear in the transmission multiplied by the gear ratio of the final drives multiplied by the circumference of the drive sprocket will equal the gear ratio limit of the tank. The tank is unable to achieve higher speeds than the gear ratio limit because it physically cannot spin its tracks around any faster.

Most spec sheets don't actually give out the transmission ratios in different gears, but such excessively detailed specification sheets are provided in Germany's Tiger Tanks by Hilary Doyle and Thomas Jentz. The gear ratios, final drive ratios, and maximum engine RPM of the Tiger II are all provided, along with a handy table of the vehicle's maximum speed in each gear. In eighth gear, the top speed is given as 41.5 KPH, but that is at an engine speed of 3000 RPM, and in reality the German tank engines were governed to less than that in order to conserve their service life. At a more realistic 2500 RPM, the mighty Tiger II would have managed 34.6 KPH.

In principle there are analogous limits for electrical and hydraulic drive components based on free speeds and stall torques, but they are a little more complicated to actually calculate.

Part of the transmission from an M4 Sherman, picture from Jeeps_Guns_Tanks' great Sherman website

The Power Limit

So a Tiger II could totally go 34.6 KPH in combat, right? Well, perhaps. And by "perhaps," I mean "lolololololol, fuck no." I defy you to find me a test report where anybody manages to get a Tiger II over 33 KPH. While the meticulous engineers of Henschel did accurately transcribe the gear ratios of the transmission and final drive accurately, and did manage to use their tape measures correctly when measuring the drive sprockets, their rosy projections of the top speed did not account for the power limit.

As a tank moves, power from the engine is wasted in various ways and so is unavailable to accelerate the tank. As the tank goes faster and faster, the magnitude of these power-wasting phenomena grows, until there is no surplus power to accelerate the tank any more. The system reaches equilibrium, and the tank maxes out at some top speed where it hits its power limit (unless, of course, the tank hits its gear ratio limit first).

The actual power available to a tank is not the same as the gross power of the motor. Some of the gross horsepower of the motor has to be directed to fans to cool the engine (except, of course, in the case of the Brayton Cycle Master Race, whose engines are almost completely self-cooling). The transmission and final drives are not perfectly efficient either, and waste a significant amount of the power flowing through them as heat. As a result of this, the actual power available at the sprocket is typically between 61% and 74% of the engine's quoted gross power.

Once the power does hit the drive sprocket, it is wasted in overcoming the friction of the tank's tracks, in churning up the ground the tank is on, and in aerodynamic drag. I have helpfully listed these in the order of decreasing importance.

The drag coefficient of a cube (which is a sufficiently accurate physical representation of a Tiger II) is .8. This, multiplied by half the fluid density of air (1.2 kg/m^3) times the velocity (9.4 m/s) squared times a rough frontal area of 3.8 by 3 meters gives a force of 483 newtons of drag. This multiplied by the velocity of the tiger II gives 4.5 kilowatts, or about six horsepower lost to drag. With the governor installed, the HL 230 could put out about 580 horsepower, which would be four hundred something horses at the sprocket, so the aerodynamic drag would be 1.5% of the total available power. Negligible. Tanks are just too slow to lose much power to aerodynamic effects.

Losses to the soil can be important, depending on the surface the tank is operating on. On a nice, hard surface like a paved road there will be minimal losses between the tank's tracks and the surface. Off-road, however, the tank's tracks will start to sink into soil or mud, and more power will be wasted in churning up the soil. If the soil is loose or boggy enough, the tank will simply sink in and be immobilized. Tanks that spread their weight out over a larger area will lose less power, and be able to traverse soft soils at higher speed. This paper from the UK shows the relationship between mean maximum pressure (MMP), and the increase in rolling resistance on various soils and sands in excruciating detail. In general, tanks with more track area, with more and bigger road wheels, and with longer track pitch will have lower MMP, and will sink into soft soils less and therefore lose less top speed.

The largest loss of power usually comes from friction within the tracks themselves. This is sometimes called rolling resistance, but this term is also used to mean other, subtly different things, so it pays to be precise. Compared to wheeled vehicles, tracked vehicles have extremely high rolling resistance, and lose a lot of power just keeping the tracks turning. Rolling resistance is generally expressed as a dimensionless coefficient, CR, which multiplied against vehicle weight gives the force of friction. This chart from R.M. Ogorkiewicz' Technology of Tanks shows experimentally determined rolling resistance coefficients for various tracked vehicles:

The rolling resistance coefficients given here show that a tracked vehicle going on ideal testing ground conditions is about as efficient as a car driving over loose gravel. It also shows that the rolling resistance increases with vehicle speed. A rough approximation of this increase in CR is given by the equation CR=A+BV, where A and B are constants and V is vehicle speed. Ogorkiewicz explains:

It should be noted that the lubricated needle bearing track joints of which he speaks were only ever used by the Germans in WWII because they were insanely complicated. Band tracks have lower rolling resistance than metal link tracks, but they really aren't practical for vehicles much above thirty tonnes. Other ways of reducing rolling resistance include using larger road wheels, omitting return rollers, and reducing track tension. Obviously, there are practical limits to these approaches.

To calculate power losses due to rolling resistance, multiply vehicle weight by CR by vehicle velocity to get power lost. The velocity at which the power lost to rolling resistance equals the power available at the sprocket is the power limit on the speed of the tank.

The Suspension Limit

The suspension limit on speed is starting to get dangerously far away from the world of spherical, frictionless horses where everything is easy to calculate using simple algebra, so I will be brief. In addition to the continents of the world not being completely comprised of paved surfaces that minimize rolling resistance, the continents of the world are also not perfectly flat. This means that in order to travel at high speed off road, tanks require some sort of suspension or else they would shake their crews into jelly. If the crew is being shaken too much to operate effectively, then it doesn't really matter if a tank has a high enough gear ratio limit or power limit to go faster. This is also particularly obnoxious because suspension performance is difficult to quantify, as it involves resonance frequencies, damping coefficients, and a bunch of other complicated shit.

Suffice it to say, then, that a very rough estimate of the ride-smoothing qualities of a tank's suspension can be made from the total travel of its road wheels:

This chart from Technology of Tanks is helpful. A more detailed discussion of the subject of tank suspension can be found here.

The Real World Rudely Intrudes

So, how useful is high top speed in a tank in messy, hard-to-mathematically-express reality? The answer might surprise you!

A Wehrmacht M.A.N. combustotron Ausf G

We'll take some whacks at everyone's favorite whipping boy; the Panther.

A US report on a captured Panther Ausf G gives its top speed on roads as an absolutely blistering 60 KPH on roads. The Soviets could only get their captured Ausf D to do 50 KPH, but compared to a Sherman, which is generally only credited with 40 KPH on roads, that's alarmingly fast.

So, would this mean that the Panther enjoyed a mobility advantage over the Sherman? Would this mean that it was better able to make quick advances and daring flanking maneuvers during a battle?

No.

In field tests the British found the panther to have lower off-road speed than a Churchill VII (the panther had a slightly busted transmission though). In the same American report that credits the Panther Ausf G with a 60 KPH top speed on roads, it was found that off road the panther was almost exactly as fast as an M4A376W, with individual Shermans slightly outpacing the big cat or lagging behind it slightly. Another US report from January 1945 states that over courses with many turns and curves, the Sherman would pull out ahead because the Sherman lost less speed negotiating corners. Clearly, the Panther's advantage in straight line speed did not translate into better mobility in any combat scenario that did not involve drag racing.

So what was going on with the Panther? How could it leave everything but light tanks in the dust on a straight highway, but be outpaced by the ponderous Churchill heavy tank in actual field tests?

Panther Ausf A tanks captured by the Soviets

A British report from 1946 on the Panther's transmission explains what's going on. The Panther's transmission had seven forward gears, but off-road it really couldn't make it out of fifth. In other words, the Panther had an extremely high gear ratio limit that allowed it exceptional speed on roads. However, the Panther's mediocre power to weight ratio (nominally 13 hp/ton for the RPM limited HL 230) meant that once the tank was off road and fighting mud, it only had a mediocre power limit. Indeed, it is a testament to the efficiency of the Panther's running gear that it could keep up with Shermans at all, since the Panther's power to weight ratio was about 20% lower than that particular variant of Sherman.

There were other factors limiting the Panther's speed in practical circumstances. The geared steering system used in the Panther had different steering radii based on what gear the Panther was in. The higher the gear, the wider the turn. In theory this was excellent, but in practice the designers chose too wide a turn radius for each gear, which meant that for any but the gentlest turns the Panther's drive would need to slow down and downshift in order to complete the turn, thus sacrificing any speed advantage his tank enjoyed.

So why would a tank be designed in such a strange fashion? The British thought that the Panther was originally designed to be much lighter, and that the transmission had never been re-designed in order to compensate. Given the weight gain that the Panther experienced early in development, this explanation seems like it may be partially true. However, when interrogated, Ernst Kniepkamp, a senior engineer in Germany's wartime tank development bureaucracy, stated that the additional gears were there simply to give the Panther a high speed on roads, because it looked good to senior generals.

So, this is the danger in evaluating tanks based on extremely simplistic performance metrics that look good on paper. They may be simple to digest and simple to calculate, but in the messy real world, they may mean simply nothing.